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We study the weak convergence of the extremes of supercritical branching Lévy processes 
$\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, 
$\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of 
$\mathbb{X}_t$.
